Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation

This article is a review of our work towards a parameter-free method for simulation of turbulent flow at high Reynolds numbers. In a series of papers we have developed a model for turbulent flow in the form of weak solutions of the Navier-Stokes equations, approximated by an adaptive finite element method, where: (i) viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and (ii) skin friction at solid walls is assumed to be negligible compared to inertial effects. The result is a computational model without empirical data, where the only model parameter is the local size of the finite element mesh. Under adaptive refinement of the mesh based on a posteriori error estimation, output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution, and thus the resulting method has no adjustable parameters. No ad hoc design of the mesh is needed, instead the mesh is optimized based on solution features, in particular no boundary layer mesh is needed. We connect the computational method to the mathematical concept of a dissipative weak solution of the Euler equations, as a model of high Reynolds number turbulent flow, and we highlight a number of benchmark problems for which the method is validated. The purpose of the article is to present the computational framework in a concise form, to report on recent progress, and to discuss open problems that are subject to ongoing research.